# Fast core pricing algorithms for path auction

- 20 Downloads

## Abstract

Path auction is held in a graph, where each edge stands for a commodity and the weight of this edge represents the prime cost. Bidders own some edges and make bids for their edges. The auctioneer needs to purchase a sequence of edges to form a path between two specific vertices. Path auction can be considered as a kind of combinatorial reverse auctions. Core-selecting mechanism is a prevalent mechanism for combinatorial auction. However, pricing in core-selecting combinatorial auction is computationally expensive, one important reason is the exponential core constraints. The same is true of path auction. To solve this computation problem, we simplify the constraint set and get the optimal set with only polynomial constraints in this paper. Based on our constraint set, we put forward two fast core pricing algorithms for the computation of bidder-Pareto-optimal core outcome. Among all the algorithms, our new algorithms have remarkable runtime performance. Finally, we validate our algorithms on real-world datasets and obtain excellent results.

## Keywords

Path auction Core Pricing algorithm Constraint set## Notes

### Acknowledgements

This paper is supported by the National Key Research and Development Program of China (Grant No. 2018YFB1403400), the National Natural Science Foundation of China (Grant No. 61876080), the Collaborative Innovation Center of Novel Software Technology and Industrialization at Nanjing University.

## References

- 1.Archer, A., & Tardos, É. (2007). Frugal path mechanisms.
*ACM Transactions on Algorithms (TALG)*,*3*(1), 3.MathSciNetzbMATHCrossRefGoogle Scholar - 2.Ausubel, L. M., & Milgrom, P. R. (2002). Ascending auctions with package bidding.
*Advances in Theoretical Economics*,*1*(1), 1–42.MathSciNetCrossRefGoogle Scholar - 3.Ausubel, L. M., Milgrom, P., et al. (2006). The lovely but lonely Vickrey auction.
*Combinatorial Auctions*,*17*, 22–26.Google Scholar - 4.Bünz, B., Seuken, S., & Lubin, B. (2015). A faster core constraint generation algorithm for combinatorial auctions. In
*Twenty-Ninth AAAI conference on artificial intelligence*(pp. 827–834).Google Scholar - 5.Bünz, B., Lubin, B., & Seuken, S. (2018). Designing core-selecting payment rules: A computational search approach. In
*Proceedings of the 2018 ACM conference on economics and computation*(pp. 109–109). ACM.Google Scholar - 6.Cheng, H., Zhang, L., Zhang, Y., Wu, J., & Wang, C. (2018). Optimal constraint collection for core-selecting path mechanism. In
*Proceedings of the 17th international conference on autonomous agents and multiagent systems*(pp. 41–49).Google Scholar - 7.Clarke, E. H. (1971). Multipart pricing of public goods.
*Public Choice*,*11*(1), 17–33.CrossRefGoogle Scholar - 8.Cramton, P. (2013). Spectrum auction design.
*Review of Industrial Organization*,*42*(2), 161–190.CrossRefGoogle Scholar - 9.Day, R., & Milgrom, P. (2013). Optimal incentives in core-selecting auctions. In
*The handbook of market design*(Chap. 11, pp. 282–298). OUP Oxford.Google Scholar - 10.Day, R., & Milgrom, P. (2008). Core-selecting package auctions.
*International Journal of Game Theory*,*36*(3–4), 393–407.MathSciNetzbMATHCrossRefGoogle Scholar - 11.Day, R. W., & Cramton, P. (2012). Quadratic core-selecting payment rules for combinatorial auctions.
*Operations Research*,*60*(3), 588–603.MathSciNetzbMATHCrossRefGoogle Scholar - 12.Day, R. W., & Raghavan, S. (2007). Fair payments for efficient allocations in public sector combinatorial auctions.
*Management Science*,*53*(9), 1389–1406.zbMATHCrossRefGoogle Scholar - 13.Du, Y., Sami, R., & Shi, Y. (2010). Path auctions with multiple edge ownership.
*Theoretical Computer Science*,*411*(1), 293–300.MathSciNetzbMATHCrossRefGoogle Scholar - 14.Elkind, E., Sahai, A., & Steiglitz, K. (2004). Frugality in path auctions. In
*Proceedings of the fifteenth annual ACM-SIAM symposium on discrete algorithms*(pp. 701–709). Society for Industrial and Applied MathematicsGoogle Scholar - 15.Erdil, A., & Klemperer, P. (2010). A new payment rule for core-selecting package auctions.
*Journal of the European Economic Association*,*8*(2–3), 537–547.CrossRefGoogle Scholar - 16.Feigenbaum, J., Papadimitriou, C., Sami, R., & Shenker, S. (2005). A BGP-based mechanism for lowest-cost routing.
*Distributed Computing*,*18*(1), 61–72.zbMATHCrossRefGoogle Scholar - 17.Grötschel, M., Lovász, L., & Schrijver, A. (1981). The ellipsoid method and its consequences in combinatorial optimization.
*Combinatorica*,*1*(2), 169–197.MathSciNetzbMATHCrossRefGoogle Scholar - 18.Groves, T., et al. (1973). Incentives in teams.
*Econometrica*,*41*(4), 617–631.MathSciNetzbMATHCrossRefGoogle Scholar - 19.Hartline, J., Immorlica, N., Khani, M.R., Lucier, B., & Niazadeh, R. (2018). Fast core pricing for rich advertising auctions. In
*Proceedings of the 2018 ACM conference on economics and computation*(pp. 111–112). ACM.Google Scholar - 20.Hershberger, J., & Suri, S. (2001). Vickrey prices and shortest paths: What is an edge worth? In
*Proceedings 42nd IEEE symposium on foundations of computer science*(pp. 252–259). IEEE.Google Scholar - 21.Karger, D., & Nikolova, E. (2006). On the expected VCG overpayment in large networks. In
*Proceedings of the 45th IEEE conference on decision and control*(pp. 2831–2836).Google Scholar - 22.Karlin, A.R., Kempe, D., & Tamir, T. (2005). Beyond VCG: Frugality of truthful mechanisms. In
*46th annual IEEE symposium on foundations of computer science (FOCS’05)*(pp. 615–626). IEEE.Google Scholar - 23.Lee, Y. T., Sidford, A., & Wong, S. C. W. (2015). A faster cutting plane method and its implications for combinatorial and convex optimization. In
*56th annual symposium on foundations of computer science*(pp. 1049–1065). IEEE.Google Scholar - 24.Lehmann, D., Oćallaghan, L. I., & Shoham, Y. (2002). Truth revelation in approximately efficient combinatorial auctions.
*Journal of the ACM (JACM)*,*49*(5), 577–602.MathSciNetzbMATHCrossRefGoogle Scholar - 25.Leskovec, J., & Krevl, A. (2014, June). SNAP datasets: Stanford large network dataset collection. http://snap.stanford.edu/data/.
- 26.Nisan, N., & Ronen, A. (1999). Algorithmic mechanism design. In
*Proceedings of the thirty-first annual ACM symposium on theory of computing*(pp. 129–140). ACM.Google Scholar - 27.Polymenakos, L., & Bertsekas, D. P. (1994). Parallel shortest path auction algorithms.
*Parallel Computing*,*20*(9), 1221–1247.MathSciNetzbMATHCrossRefGoogle Scholar - 28.Rothkopf, M. H., Pekeč, A., & Harstad, R. M. (1998). Computationally manageable combinational auctions.
*Management science*,*44*(8), 1131–1147.zbMATHCrossRefGoogle Scholar - 29.Vickrey, W. (1961). Counterspeculation, auctions, and competitive sealed tenders.
*The Journal of Finance*,*16*(1), 8–37.MathSciNetCrossRefGoogle Scholar - 30.Yokoo, M., Sakurai, Y., & Matsubara, S. (2004). The effect of false-name bids in combinatorial auctions: New fraud in internet auctions.
*Games and Economic Behavior*,*46*(1), 174–188.MathSciNetzbMATHCrossRefGoogle Scholar - 31.Zhang, L., Chen, H., Wu, J., Wang, C. J., & Xie, J. (2016). False-name-proof mechanisms for path auctions in social networks. In
*ECAI*(pp. 1485–1492).Google Scholar - 32.Zhu, Y., Li, B., Fu, H., & Li, Z. (2014). Core-selecting secondary spectrum auctions.
*IEEE Journal on Selected Areas in Communications*,*32*(11), 2268–2279.CrossRefGoogle Scholar